Explicit solutions to dynamic diffusion-type equations and their time integrals

Abstract

This paper deals with solutions of diffusion-type partial dynamic equations on discrete-space domains. We provide two methods for finding explicit solutions, examine their asymptotic behavior and time integrability.

These properties depend significantly not only on the underlying time structure but also on the dimension and symmetry of the problem. Throughout the paper, the results are interpreted in the context of random walks and related stochastic processes.

Keywords

• Diffusion equation
• Heat equation
• Semidiscrete equations
• Partial dynamic equations
• Time scales
• Random walk